Given an undirected graph $G=(V,E)$ a Gomory-Hu tree $T$ for $G$ has the following properties:

  • $T$ has a node for each vertex in the graph G and each edge in the tree corresponds to a minimum cut separating the corresponding pair of vertices of G.
  • $T$ has the property that the minimum cut between vertices $s$ and $t$ in $G$ is the cut corresponding to the minimum weight edge along the path from $s$ to $t$ in the Gomory-Hu tree.

In $[1]$, the author states the following lemma:

Let $S$ be the union of cuts in $G$ associated with $n$ edges of $T$. Then removal of $S$ from $G$ leaves a graph with at least $n+1$ components.

Consider a complete graph $K_4$ with all the edges having weight $1$.
A (possible) Gomory-Hu Tree $T$ for $K_4$ is the following:

If we remove the cuts associated with all the $n=3$ edges of $T$ we might proceed as follows:

  • $(3,0) \in T$ $\rightarrow$ remove $\{(3,0),(3,1),(3,2)\} \in G$
  • $(3,1) \in T$ $\rightarrow$ remove $\{(3,0),(3,1),(3,2)\} \in G$
  • $(3,2) \in T$ $\rightarrow$ remove $\{(3,0),(3,1),(3,2)\} \in G$

A potential result (among the different ones) is a graph with $2 \neq (n+1=4)$ connected components, i.e., $\{3\}$ and $\{1,2,4\}$.
So, my question is, am I missing something in the definition or properties of the Gomory-Hu trees?
Does the author mean that for each $(u,v)$ in $T$, all the minimum weight cuts between $u$ and $v$ in $G$ should be removed in order for the lemma to hold?

[1] Vazirani, Vijay V. Approximation algorithms. Springer Science & Business Media, 2013

  • $\begingroup$ When you are saying "union of cuts", does it mean "union of cut sets"? Meaning of Cut-set: the set of edges that goes across the cut. $\endgroup$ – Inuyasha Yagami Jan 10 at 20:13

The book states the following definition of a cut:

a cut is defined by a partition of $V$ into two sets say $V^{l}$ and $V \setminus V^{l}$ and consists of all edges that have one endpoint in each partition.

Therefore, removing a cut from the graph should mean that we are removing the edges that go across the cut.

Also, consider the following definition of the "cut associated with an edge $(u,v) \in T$ in $G$".

Each edge $(u,v)$ in $T$ denotes a partition of $V$ given by the two connected components obtained by removing $(u,v)$ from $T$. Consider the cut defined in $G$ by this partition. We will say that this is the cut associated with $(u,v)$ in $G$.

Now, let us consider your example of Gomory-Hu tree for $K_{4}$. We want to find the cut associated with the edge $(3,0) \in T$. If we remove $(3,0)$ from $T$, we obtain the partitioning $\{1,2,3\}$ and $\{0\}$. This defines the cut in $G$. Therefore, you should be removing the edges of $G$ that goes across this cut i.e., $(0,1)$ $(0,2)$ and $(0,3)$.

Similarly for the edge $(3,1) \in T$, you should be removing the edges: $(1,0)$, $(1,2)$, and $(1,3)$.

And, for the edge $(3,2) \in T$, you should be removing the edges: $(2,0)$, $(2,1)$, and $(2,3)$.

It means we are removing all edges from the graph. Therefore, the number of connected components in the remaning graph would be $4 = (n + 1)$, for $n = 3$.

  • 1
    $\begingroup$ Nice, I had totally missed the definition of cut associated with (𝑢,𝑣) in 𝐺. $\endgroup$ – abc Jan 10 at 20:47

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